2
votes
1answer
327 views

Are all linear transformations measurable?

Let $V$ and $W$ be topological vector spaces over $\mathbb{F}$ (with $\mathbb{F}=\mathbb{R}$ or $\mathbb{C}$), and let $T:V \to W$ be a linear transformation. It is well-known that $T$ is not ...
4
votes
0answers
153 views

When is the sum of a weak-$*$ closed convex cone and a subspace also weak-$*$ closed?

Let $X$ be a Banach space. Suppose $C \subset X^*$ is a convex cone and $V \subset X^*$ is a subspace, and suppose both $C$ and $V$ are closed in the weak-$*$ topology. Are there any general ...
2
votes
0answers
64 views

Are there pathological examples of log-concave measures that admit no shifts?

Does there exist a random vector $X$ in, say, the space $\mathbb{R}^\infty$ of sequences that has the following properties? The distribution of $X$ is log-concave, i.e. for every $n$ the joint ...
-1
votes
1answer
85 views

Exponential Convexity Results [closed]

$\textbf{Definition:}$ 1. A function $h : (a,b)\rightarrow\mathbb{R}$ is exponentially convex if it is continuous and $$\sum _{i, j=1}^n\xi_i\xi_jh(x_i+x_j)\geq 0,$$ for all $n\in\mathbb{N}$ and all ...
0
votes
1answer
132 views

Exponential Convexity

$\textbf{Definition:}$ 1. A function $h : (a,b)\rightarrow\mathbb{R}$ is exponentially convex if it is continuous and $$\sum _{i, j=1}^n\xi_i\xi_jh(x_i+x_j)\geq 0,$$ for all $n\in\mathbb{N}$ and all ...
0
votes
0answers
65 views

Open ideally convex sets

Background Recall that a subset $A \subseteq X$ of a Banach$^{1}$ space $X$ is said to be ideally convex if, for every bounded sequence $(x_n)_{n \in {\mathbb N}}$ in $A$ and every sequence ...
0
votes
0answers
100 views

Uniform convergence of difference quotients of a convex function

Let $f(\cdot):\mathbb{R}^n\rightarrow \mathbb{R}$ be a convex function. At $x$ denote the subdifferential by $\partial f(x)$ which is compact and closed. Now, define the approximation of $f$ around a ...
0
votes
1answer
558 views

Is a convex function continuous and almost everywhere differentiable? [closed]

is this statement true ? assume $f:D\rightarrow \mathbf{R} $ is a convex function where $D\subset \mathbf{R}^n$ is a convex set. $f$ is continuous and almost everywhere differentiable and in class ...
3
votes
1answer
99 views

What is the doubling dimension of convex functions?

I am interested in the complexity of convex functions, specifically the "doubling dimension" of the class of convex functions defined on a compact subset of Euclidean space, when compared using the ...
6
votes
0answers
293 views

Minkowski's Inequality for Integrals in Orlicz spaces

EDIT: I have changed the question to have less parameters, fitting it into the context of Orlicz spaces. Suppose $f:[0,\infty)\to[0,\infty)$ is convex and increasing, ...
7
votes
2answers
267 views

Duality between extremal points and extremal maps

Suppose I have a convex set $C\subset\mathbb{R}^n$ such that $0\in C$ and every Cauchy sequence in $C$ converges in $C$, but $C$ need not be bounded. (Actually I want unbounded $C$). Consider the set ...
3
votes
1answer
167 views

Are faces of a compact, convex body “opposed” iff their extreme points are pairwise “opposed”?

Let $P$ be a compact, convex subset of $\mathbb{R}^n$ (infinite-dimensional generalisations welcome, but not necessary). Let's say that disjoint subsets $W_1$, $W_2$ $\subset P$ are opposed if there ...
13
votes
2answers
466 views

A natural center of a convex weakly compact set in Banach space

Question: Let $S$ be a convex weakly compact set in Banach space $H$. Propose a natural way to define the unique center $O \in S$. Motivation: A lot! For example, in game theory $S$ can be a set of ...
13
votes
2answers
451 views

Extreme points of unit ball in tensor product of spaces

Let $B_1, B_2$ be unit balls in finite-dimensional normed spaces $X_1, X_2$ respectively. Let $e(B_1), e(B_2)$ be corresponding extreme points sets. Consider the unit ball $B$ in tensor product ...
1
vote
0answers
106 views

Mappings preserving convex compactness

Let $H$ be a Hilbert space. How can one describe continuous mappings $F:H \to H$ that satisfy the following condition: There exist two elements $c$, $F(c) \neq c$ and a convex compact $M$ containing ...
5
votes
3answers
1k views

A criterion for the sum of two closed sets to be closed ?

Let $V$ and $I$ be two closed subsets of a Banach space $A$. The set $V$ is a convex cone, and $I$ is a linear subspace of $A$. I also know that $V\cap I=\{0\}$. I would like to know whether $I+V$ ...
21
votes
1answer
461 views

The maximal “nearly convex” function

The following problem is only tangently related to my present work, and I do not have any applications. However, I am curious to know the solution -- or even to see a lack thereof, indicating that the ...
5
votes
0answers
258 views

Local minimum from directional derivatives in the space of convex bodies

I have a function $f(K)$ defined on the space of three-dimensional convex bodies for which I want to show that the unit ball $B$ is a local minimum. I have been able to show if $K$ is not homothetic ...
11
votes
1answer
419 views

Status of the compact AR problem?

The so-called "compact AR Problem" reads: Is every compact convex set in a metrizable topological vector space an absolute retract? It is open according to the chapter by T. Banakh, R. Cauty ...
0
votes
0answers
315 views

Global index of convexity/concavity of a function

We are looking for a global index of the convexity/concavity of a function. For concreteness, how can I formalize the intuitive notion that a function $f$ is more convex than $g$ where ...
2
votes
0answers
232 views

Are affine continuous functions on Bauer sub-simplices of the probability measures given by integration over continuous functions?

Let $X$ be a compact (non-metrizable) Hausdorff space and $\mathcal{P}(X)$ the set of Radon probability measures with weak-$*$ topology (weak topology induced by the continuous functions). Consider a ...
0
votes
1answer
236 views

Integral in a σ−convex set.

Having had no (proper) answer to this question, I formulate the remaining case as a new question as follows. With $I=[0,1]$, let $E$ be a separable (real) Banach space, and let $\gamma:I\to E$ be ...
8
votes
1answer
557 views

volume of the unit ball of the Banach space $\ell_1^n\otimes_{\epsilon}\ell_1^n$?

We denote by $\otimes_{\epsilon}$ the injective Banach tensor product. Which is the asymptotic volume of the unit ball of the Banach space $\ell_1^n\otimes_{\epsilon}\ell_1^n$?
2
votes
2answers
430 views

Existence of extreme points

Let $(X,d)$ be a complete metric linear space whose balls are convex. Let $Y\subseteq X$ be a bounded, closed and convex subset that verifies the following property: for all $y_0\in Y$, the distance ...
12
votes
0answers
1k views

Infinite convex combinations in a Banach space

Let's say that a subset $C$ of a Banach space $X$ is $\sigma$-convex if the following property holds: For any sequence $(x_k)_{k\ge0}$ in $C$, and for any sequence of non-negative real numbers ...
1
vote
0answers
340 views

Bounding point-wise maximum of the absolute difference of two convex functions

Let $\Delta: R \times R \rightarrow R_{+}$ be a positive and convex function (convex in, say, both the arguments) called the loss function. Let $x \in R^d$. Moreover, let $H_1,...,H_r$ be sets of ...
5
votes
1answer
559 views

Does a log-concave function on a convex set extend continuously to the boundary?

Let $U$ be an open convex set in a locally convex space $X$, and let $f : U \to [0,1]$ be a log-concave function on $U$ (i.e., bounded and real-valued). Under what conditions does $f$ have a ...
1
vote
0answers
359 views

Does the dual Banach space $B(\ell^\infty)$ has weak* normal structure?

Let $K$ be a bounded closed convex subset of a Banach space $E$. A point $x$ in $K$ is called a diametral point if $$ \sup_{y\in K} ||x-y||=diam(K). $$ where $diam(K)$ denotes the diameter of $K$. ...
1
vote
0answers
127 views

Lower semicontinuity of Bregman distances/divergences

For a Banach space $X$ and a convex functional $J:X \to [0,\infty]$ (i.e. with values in the extended reals), consider the associated Bregman distance: For $x,y\in X$ and $\xi\in\partial J(y)$: ...
3
votes
1answer
648 views

Lipschitz properties of minima/minimizers of convex functions of two variables

Suppose I have a function $f(x,y)$ from $\mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ that is convex in both $x$ and $y$. Set $g(y) = \min_{x} f(x,y)$ What I would like is for $g(y)$ to be ...
0
votes
1answer
285 views

Convex sets and projections

Hello! I recently started (it's purely self-education) reading a "Mathematical programming and optimizations" book, did a vast part of the exercises related to the theoretical part and at one moment ...
1
vote
6answers
3k views

Extreme point compact convex set.

The Krein-Milman theorem shows that a compact convex set in a Hausdorff locally convex topological vector space is the convex hull of its extreme points. It seems this implies that a compact convex ...
4
votes
2answers
2k views

Compact Convex sets and Extreme Points

There are examples that show the set of extreme points of a compact convex subset of a locally convex topological vector space need not be closed when the real dimension of the space is at least 3. ...